SCIENTIFIC ABSTRACT GODUNOV, S.K.  GODYCKICWIRKO, T.
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CIARDP8600513R0006155200129
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December 31, 1967
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SCIENTIFIC ABSTRACT
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Difference Methods for the Numerical Calculat4Lon
of the Discontinuoas Solutions of llydrodynami,~ wiplaionS
?U = 1) V
system A ~V B a'11 fzr automatic calcu.
57 , 
6 Ux jt IT x
equipments the f3llowing scheme ig rocommended
A
u u + (v V1) V VVAB (U i U
o 2h I 2h 0
Va V + u (V 2 +
0 2h '1 21,
In the case of acol;stic waves the auLhor giros an interesting
physical interpretation for (1) whih then is used in Chaper
II in order to obtain the differenc,~! scheme for platie aridl
mensional instatioriary hydromechac,',(, L&Krange Pquat,c~rzr.1
2
eii (E +
T
du JD(V,E) L11 2 al)u
= 0 0 t + B di
5t 3 x F) t ax
Card 213
Difference Methods for the Nwmerl._al Cair~ulatior, 5 0 71/ 3At 72 / 41
of the Discontinucus Solutions of 'llydrodynanic Fpations
Since the system can possess nonsmor.th solul'ions ;ver, 1'~;r
continuous inittal conditiois, gerer%'.'L,ed 8oJ,.,~if:),,s (duo to
Sobolevt vith ixpazt waves ALre incluided in the ~onsiderutiozn~
It is Pro7eLl 'that the propoeed shome urder con,;ergi~ncc tende
to these genprali2)d snlutions, NumerouL~ t)~ne 4
,!a p~oppT~'es
of t'l.e scheme and experierces calculatiOrl 41'P [';17(3~ Tt,e
scheme I's used t,,y Soviet elf%_~tronin Vnt~ e:jt~i.r re,
marks that ni.mllar izthodq hive bec,r. 2e,~P'Loped NN111.
Yanenko. He furthi~rmore menticns LA. Sediv ar.!
There are 7 flgU~e3 avid A referen~(ls, 2 !* ithl,.li a,F.
L) U Dn.L L L ~ij 1  1956
Card 3/3
GODUNOV, S. K.) ZITUKOV) A. I., S~XMDYEV, K. A. Omoscuv)
"Numerical Methods in the Analysis of OneDimenBional Unsteady Problems of
Gas Dynamics."
report presented at the First AllUnion Congress on Theoretical and Applied
Mechanics, Moscow, 27 Jaa  3 Feb 1960.
84659
4.SSvu 16 lliao S/02 6o/134/006/'002/031
CI 1 IYC222
AUTHORt Godunov, SA,
TITLEs On the Concept of Generalized Solution
PERIODICALs Doklady Akademii nauk 3SSR, 1960s Vol. 134, Ko. 6,
pp. 1279  1282
TEXTt The author considers the quasilinear hyperbolicaystem_14
gFi(qllq 2f .Iq n)
(1) ?t
?Gi(qllq 2"qn)
'?X
. 0
Generalized solutions are those q,(X,t), q,(X,t),.,q,(x,t), for which
on each contour it hc;lds
0 1dx  Gi dt  0
However, it is senseless to denote all functions which satisfy thi3 con
Card 1/2
84659
On the Concept of Generalized Solution S/020/60/134/006jOO2/031
C111/022
dition as solutions. I.M. Gellfand (Ref. 1) proposed to denote those among
the q, as solutions which one obtains as limit values for 0 from
the solutions of
(2) 9Fi ~Gi F, b '?qk
? t ~ x  ~x ik 4 X
Such a definition would have a sense if it would be fixed uniquely by (1)
and if the matrix 11 bik 11 would not influence the result of the limiting
process.
By an example the alathor shows thatthis is not the case. The fact that (1)
is hyperbolic is not sufficient in order to guarantee the uniqueness of the
mentioned limiting process (its independenoe of 11b ik 11)"
There in 1 figure and 1 Soviet reference.
SUBMITTEDi May 30, 196o
Card 2/2
GODUNOVS S.K. (Pbskva)
Evaluation of the inaccuraciee occurring in the derivation of
approximate solutions to simple equations of gas dynamic3. Zhur.
vych.mat.i mat.fiz. 1 no.4:622637 JIAg 161. (KMA 14:8)
(Approximate computation) (Differential equationB)
(Gas dynamics)
ILI
0/208/61/Al/006/005/013
Ill 12/B) 38
AUTHORS: Godunov, S. K., Zabrodin, A. V., Prokopov, G. P. 'Moscow)
TITLE: Difference scheme for twodimensional nonstationary
problems of gas dynamics and calculation of a flor. with a
shock wave that runs backward
PL'R10DICAL: Zhurnal vychislitellnoy Liatematiki i itatematicheaVoy fiziki,
v. 1, no. 6, 1961, 102010r0
TEXT: In this paper, the authors continue investigations of difference
schemes for nonstationary problems of Cas dynamic3 (cf. S. K. Godunov.
,.;atem. sb., 1959, 47, no. 3. 271306). in order to solve the sysilem
Pd,1,dY+P11dY(11! Pl.drdt o.
pu dx dy + (p P pu) dy dl ~ pu v dr dl 0, (2.2)
pr dx dy  ~ ptiv dy (11 + (p + pO) dx (11 0,
Card 1/3
Difference scheme for twodimensional... B112/B138
U2 + III
p (e + !~~) dx. dy + pit (e + + dy fit
2 (2. 2
+ L' 2_1 dx (it 0
+ Pv ( 1, 2
the authors use the following difference ocheme
It3 Is n 3i2
is
n
tit I 3~
If
1113 1,2 n111, n +31:, hv
111+112 nt + Ih Pi2
_P hx
Card 2/3
31108
S/208/61/001/036/005/013
Difference scheme for twodimensional ... B112/B138
Discontinuity disintegration is calculated using the scheme
P~ + P,,+',,, N11, + Pn+".
a. b. VTrE 2 2
Pn4.t/, + PV, a. UnPI, Un (3.3)
PH. P. 2
U n+". + Uny'+Pns +r 1111  P.1
2 2an
It is based on the formula p  (7  1 ?e. The stability condition of the
scheme is derived. In the latter part of the article, the auth)rs use
nets which are moved in accordance with the flow. Cases of axial
symmetry, in particular that of a sphere, are considered.
I. G. Petrovskiy, 0. M. Beloteerkovskiy (Prikl. matem. i mekhan., 1960,
24, no. 3, 511517), and A. A. Dorodnitsin are mentioned. I. Lt. Gellfand,
K. A. Bagrinovskiy, G. N. Novozhilov, V. V. Lutsikovich, and
K. A. Semendayev are thanked for assistance. There are 15 figures and
3 Soviet references.
SUBMIITTEDt May 7, 1961
Card 3/3
3/04 2/61/016/00 3/00 4/'~,O 5
LG _ILI C 111 /C444
AUTHOR: Godunov S. K.
TITLEs On the numerical solution of boundary value Probliams for
systems of linear ordinary differential equatioris.
PERIODICAL: Uspekhi matematicheakikh nauk, v.16, no.5, 1961, 171174.
TEXT: Proposed is a numerical method for the solutlon of bcundark
value problems.
y A(jc)y + f(x)
13Y(0 0, C~(1)  0
where y,f are vectors and A,B,C are matrices. The inteval (col is
divided by points 0  x 0 e x1 4 X 2"' < x n 1. M S z 0 (XS ) be the
result of the integr..tion of the system yt Ay + f with the initial
conditions y(X,)  Z O(X8 ) from x  x 8 to x x,+i* Let z i(xo)  yi(O)
(j  0)112,...,k). If integrating the equations from x 0 to x, one gets
n j (x, ) . 140 z j (x 0) (1  0,1,29#.tyk)
The vectors n,(X,), n2 (x,), .... I nk (x1) have to be orthogonalized
Card 1/ 5
3/042/61/016/003/004/005
On the numerical solution ... C111/C444
and normed. The obtained vectors be zi(xi), Z2 (x dt ... Zk(xl) The
orthogonalisation formulas are
(Oil V1,11. Ud.
zi U,
(all
W21 (,,I. ZI). (022
Z2 t(U2  W21zo;
W22
6)31 ~ 03 Zl) (022  (U3 Z01 W33 OUS, U3) O)Tn
Z3 M L (113  6)3121  (112122);
W33
......... ...................
.............................
L okazl  (~,Szii
Whit
Wol  (118. zi), C002 0" (116, Z2), WdA,
zo  Ito  (001ZI  W0222 wohzh.
Card 2/ 5
7 5
S1042167101610031)341005
On the numerical solution ... CIIIJC444
The orthogonalioation is noted in general by Np by N in the point
N(a)
X8. By application of
one gains the triangular
W(Ij
I
1. 2 Wl.'2
1. 3
. . . . . . . . . . . . . . . . . . . .
0 "it") h  I
(D~k)
matrix.
w(4
oil
WV2
6111
. . . .
W~?h 
WW
By aid of integrations and orthogonalioutions the following sequence
is constructed.
XN N (2)
z (XI) M 0 z j (X 0) z, (x2) U (XI)
,(n) x 9 Oplp2t ... , Ic
zj (xn)
Every solution of y'  Ay + f, satisfying the boundary conditions on
..C~ard.3/5
21175
9/042 61710i6/003/004/005
On the numerical solution C111YC444 (n)
the left end, will get the value y(i)  z 0 (xn) + j zj(Xa
J, t
on the right end. The coefficients 0 (n) are obtained from the system
j
Gy (1)  0. The calculation of the solution in x of a  0,1,2,...,n
follows by formula y(X a) 0 Z n(Xs) + r' 0 00 zj(x.), whsre is
jai j
recurrently obtained from 3(8+1) by aid of the matrix
Let Oka)
69)
(a). 0.2
then the recurrence formula is CI(S+1) 11 (S) ($+I)
The author thanks I. A. Adamskays. and 1. E. Shnoll for the nutmerical
Card 4/5
On on th, sa!
al tr.,j fo r di
Thti~, :,*v )vi,,, T)l r r
S~IPW,T TTED 0  tob,
'AD V0011/)04/00 I
C
c a. I
89599
8/020/61/136/oO2/002/034
0 111/ 0 333
AUTHORs Godunov,S.Xl
TITLEs No Unique "Blurring" of Discontinuities in Solutions to
Quasilinear Systems
PERIODICAL: Doklady Akademii nauk SSSR, 1961, Vol. 136, No. 2,
pp. 272273
TIM The author considers differential equatione which describe
the "blurring" of the discontinuities in quasilinear hyperbolic
systems. In order that a "blurring" takes place, the right sides
of
; Fi(q19 q29 ... P qn) G,(ql# q29"40 qn)
0) 9t a x . 0
must be replaced by the viscosity terms 'a (E' 2 b 
,a xt  06 X ik VIA
If one seeks solutions of the form q,  qj ( 15 2.)qj(r,),
then one obtains ordinary differential equations (see (Ref.1)).
The author shows that reasonable 1b 11 can exist for which the
solution describing the "blurred" disAntinuity .1o not unique.
Card 1/3
89599
S/020/61/136/002/CO2/034
C 111/ C 333 ,
No Unique "Blurring" of Discontinuities in Solutions to Quasilinear
Systems
The system
aL L1
(2) q, qZ q b qk
+ 79 76 X ik 79 x
with
L q2+ 3q2+ 5q2 + 2e q, + 48qa. + 6eqj
1 2 3
L 1 q2 + q2 + q2 + eq, + eq% + e qj
1 2 3
is hyperbolic for &  0. In order that (2) with E.> 0 be an evolution
system in linear approximation it is sufficient that 11b 11 is
positive definite. The equations for solutions q,  qi (J~ of (2) aret
(3) Aqi _42 D ;
Card 2/3 dqi  6'1d Tr
89599
S/020/61/136/002/002/034
C 111/ C 333
No Unique "Blurring" of Discontinuities in Solutions to Quasilinear
SKstems
w ere
A. L oLL 1 A1q1  A q  A q ; D b 1i are integra
2 2 3 3 2 ik 61 CIO
tion constants. The trajectories defined in the space (.q,., q2t q3)
by (3) are orthogonal in the sense of the metric D to the equi
potential surfaces of the functionA.To the "blurred" discontinui
ties there correspond q,(T) which tend to finite boundary values
for T" 4 t oo which are tationary points of the function
A(ql;lq2 q,); The author Puts a4  7/2, A,  3/2, A  0 and
A 7 2,'an hows by topological investigation of tie equipoten
t1a surfaces and of the structure of certain critical points
that under variation of the metric D one can attain a nonuniqueness
of the trajectories which describe the "blurred" discont:~nuity.
There are 2 figures, and 1 Soviet reference.
[Abstracter's notes (Ref.1) is a paper of J. M. Gellfand In
Uspekhi matematicheskikh nauk, 1959, Vol. 14t No. 2.3.
PRESENTEDt June 30, 1960, by M. V. Keldyah, Academician
SUBMITTEDs June 21, 1960
Card 3/3
GODUNOV, S.K.
Instance of nonuniquenoss for a ncnlinear parabolic system.
DAL P11 SSSR 136 no.6:12811282 F 161. WIRA 14:3)
1. Predstavleno akademikom 1. G. Petrovskim.
(Differential equations)
J60
tTHORt
TITLEs
Godunov, S.K.
25702
S/02o/61/139/003/001/025
C111/C222
An interesting class of quasilinear systems
PERIODICAL: Akademiya nauk SSSR. Doklady, v 139, no. 3,1961, 1.21523
TEXT: The author points to a class of differential equations including
a number of essential equations of mathematical physics and being suitable
for the foundation of a mathematical theory.
The equations of the reversible processes belonging to this class have
the form ?Lqi DLJ
q
t 7xil . 0
j
where L  L(q,, q 2" ..,qn Li . Lj(ql. q29 ... v qn).
The following equations lead to this class t
1. Variation equations of Lagrange
Card 1 /S
An interesting class ...
I
25702 \J
S/020/61/139/003/001/025
C111/C222
~ ( 3 te D ~ i ( 9,f )
U(k) ( ),;M)  + T2 (k)
t x 1 X2
'(1 UO U(2) j...9 U
t x1 x2 t x 2
For a reduction put
q U(k) q q
3k t 3k1 3k2
;U
xi i2
L (k) + U(k) L1 u(k) L2. u(k) X
x x t t 1,,7k7
k IU 1 2 ;u77 k TUM k u
x1 x 2 1 x1 x2
2. Differential equations of crystal optics.
3. Equations of gas dynamics
Card 21S
An.interesting class
25702
S/020/61/139/003/001/025
C111 C222
Oput + J (PU', + P) JPUIUI + CIPUI43
+
dPU2
+ Oputul + a (Put' + P) OPUSUS
+ 0
1
opus
1i + Opusul Opusut (pUS2 + p)
+ 0,
TXT Tx
+
.
OP+ apul + opus + Opu. .0



,
di ax, Tx
rxs
2
as'
U2 U2 III + U 12 + "31
6p (E + I +
+ 2 Opul + P +
2
+  7 +
all
u'
aput E +. P .4 u" + "I aPAS, E + .~L + u1, + U1, + U.,
21
P
+
O + P
x, Oxe
For a reductton put
Card 31S
~~Pflwxtljfitr,. 1412511141, 1111THEII 1111411 MI IM41111i Ilia III llitillitilillillilHigigilliLFIFAHEII�iiulmli
25702
An interesting class ... S/02 61/139/003/091/1)25
C1 1 IYC222
U U 2 u3
q
'11 T 2
T
2 2
U + U + U2
1 2 3
E+ R

_ 2
S
q
4 S
T
U
JP 2
L L 
L
U
2p
U
3
L
9
T T
The systems (1) can be written in the form
.;qk 7
hk
J
L q,q 7t + Z L
q q
k
k i k 0
J,
k
wherefrom it follows that on a convex L ( ',g q
n) thiy &ro a natural
qlg q2t'
Card 415 ,
25702
S/02 61/139/003/0()1/025
An interesting class as. C111YC222
nonlinear generalization of the symmetrical systems of X.0s. Fziedrichs
(Ref. 1 s Comm. on pure and Appl. Math., 7, no. 2 (1954)). Tho correctness
of the system (1) can be proved with the aid of the energy inegrals for
the derivatives.
The equations of the irreversible systems can be obtained front (1) by
adding of dissipat ve terms
PL ILI
q, + q i q 1 (3)
1~ t b a
j Txj J,k,s j ' k 4'x8
the matrix It b iBilis symmetrical in J,s and i,k, furthermore it is positive
ik
definite. The symmetry follows from the conditions of Onsager for irre
versible processes.
The author mentions I.G. Petrovskiy. There are 6 Sovietbloc and 3 non
Sovietbloc references. The reference to the Englishlanguage publication
reads as follows s K.O. Friedrichs, Comm. on Pure and Appl. Math., 7,
no. 2 (1954).
PRESENTEDs March 17, 1961, by I.G. Petrovskiy, Academician
SUBMITTEDs March 7, 1961
Card 5/5
GODUNOV, S.K.
[Difference methods of tsolving gas dynamics, aquati,rit";
Raznostr.ye metody reahenlia uravuenil gazovol di=dki;
lektsii dlia studentov NGU. Novosibirsk, Novoslbirsk~i
gos. univ., 1962. 96 p. (MVIRA 17:8)
PHASE I BOCK EXFLOITATION SoV16,11011
Qq_qunov, Sergey Konstantinovich, an, Viktor Solomonovich Ryabenlkly
Vvedeniyev teoriyu, raznostnykh skhem (Introduction to the Theory
of Difference Schemes) Moscow, Fizmatgiz, 1962. 340 p.
10,000 copies printed.
Ed.: G. I. Biryuk; Tech. Ed.: L. Yu. Plaksh.
PURPOSE: This book is Intended for mathematicians who have to
solve partial differential equations and for students 6f the
third and more advanced university courses. The introduction
arid chapter I are intended for less qualified readers and may be
used in the training of technicians in computation.
COVERAGE: This book develops the concepts and techniques used In
the solution of differential equations by finitedifference
methods. It covers basic theory of difference equatIons,
convergence of their solutions to the solution of differential
Card 0
Introdliction to the Theory (Cont. )
so v16 4
(~quatlons, stability of difference schemes, the order if
approximation, the application of finitedifference sc*,,Iemes to
partial differential equations, and the stability of clLfferew!e
ochemes applied to the solution of equations of nonst"atlonary
processes by use of the spectral theory of difference
~o personalities are mentioned. There are 45 referenc~,s:
37 Soviet (including 2 translationa, 1 from the Engliail, I fromi
the German), 5 English, and 3 German. The appendices are
accompanied by 23 references: 14 Soviet, 8 English, and 1
German.
TABLE OF CONTENTS:
Preface
Introduction
)b
Card 2/9
_`267
S/208 62/002/001/001/016
GS00 D299%303
AUTHORS: Godunov, S.K., and Semendyayev# K.A. (Yoscovl)
TITLE: Difference methods for the numerical solu*.ion of
gasdynamic problems
PERIODICAL: Zhurnal vychislitellnoy matematiki i matematicheskoy
fiziki, v. 2, no. 1, 1962, 3  14
TEXT: Various numerical methods nnd 'their range of applicability
are considered and some unsolved problems are discussed. in case of
moving singularities, it is convenient to use moving grids. connec
ted with the singularities; thereby it becomes unnecessary to arti
ficially introduce independent variables~ For onedimens:.onal prob
lems. Lagrangian coordinategrids are more suitable in this respect
(than Eulerian). Moving grids are used in onedimensiona1 problems
involving contact discontinuities and in unsteadyflow problems
past cylindrical bodies. A particular type olf moving grid (for one
dimensional problems), is the one formed by 2 famillei3 of charac
teristics, llowever~ the method of ch.!.racteriotics i'~ tiot saticifac
Curd 1/5
~ 5 L, r~, 7
31208162,100211'001,"001 /016
Difference metliods for the ... D299/D303
tory, because it. does not adequately take into acc,~Iunt the smooth
ness of the soughtfor functions. The authors develop~,O a computa
tional method, whereby the Crid of the 3 families of characteria
tics is associated with the straight lines t ~ const. Thi.3 merhod
however, was not further elaboiated as it cannot be extended to the
general equations of state, Witil regard to the var;.ou~i di2ference
schemes, by which the gasdynanacs equations are approxima.ed, the
optimization problem (i.e, how tc obtain results of the desired de
gree of accuracy with the least amount of comDutational work) has
been quite insufficiently studied in technical literature. Further,
the criteria are discussed for the choice of variables, As an exam.
ple, a difference scheme is considered for CaICL1111ting a centered
expansion wave in the (abovementioned) grid, i.e, the lines t z
const. and the fanily of characteristics issuirif, froin, the cen
ter, The possibilities inherent in the use of ciettro,,.i.c t:cmputers
for solving gasdynamics problems are considered. .S 1.'Iethods
of continued calculation. These met ,hods involve .uOtIon 0."
"viscosity" in the differential _'(jt,v.tions,. It as, rv!1,,,,rt,;6 1hut the
equation of state is convex (i.e, the Bethe'Jeyl cond:Ltior. is sa
Card 2/5
13 2.r ,t
S/208/62/002/001/001/016
Difference methods for the ... D299/D303
tisfied). In this case it can be assumed that a unique e,,eneraliZed
solution exists, although this is not proved. Thereby the main dif
ficulty is the possible accumulation of singularities., The charac
ter of such an accumulation was neither studied by purely ',lathema
tical methods, nor by applied methods. In this connection, the pos
sible continuation of the solution (through the sin,,ularilies which
viere smoothed), deserves particular attention. Further, the conver.
gence of the series solution is considered. The authors made an
experimental calculation of expansion waves in a loc..,I secondorder
of accuracy scheme. Thereby it was found that the oHe.s of weak
and of strong convergence coincided. It is noted that for the fur
ther development of computational methods based on the use of the
generalized solution, it is necess,~ry to first render more exact
the latter concept. It is also noted that expansion waves are not
dealt with in literature concerned with methods of continned calcu
lation, although difference methods yield partLeularly inadequate
convergence for expansion waves. The above considerations regard
ing the onedimensional problemp fully apply to inult.dim.nisional
problems, too. The numerical methods should be baoed on t`ie concept
Card 3/5
3 3 2!'7
S120 62/002/001/001/016
Difference methods for the ... D299%303
of general solution, but should at the same tirne make all)viance
for the rough structure of the solution. With rc,!~ipect to tA,.c, 'ch:)j
ce of the grids and variables, A.A. Dorodnits.na's me"Whod of Inte
gral relationships is recommended (Ref. 11: O.M. Be~lotserkovskiy,
Raschet obtekaniya krugovogo tsilindra s otoshedshey udarnoy vol
noy. Sb. "Vychisl. matem.", M., Izdvo AS SSSR, 1959, no. 3, 14"1)
185). This method yields a high order of accuracy, using 23 cor_,Ipu
tation points only. Purther, the advantages and disadvantages of
explicit and implicit difference schemes are considered. The rel
tion between the steady and unsteady flowproblems L~j discussed.
(Above, unsteady problems were considered). A flexibl~ method is
proposed, whereby the suitable variables can. be .telected (in the
difference scheme), irrespective o 'f the equatione of state, This is
achieved by using a separate (,general) subprorram for the equa
tions of stated 1n conclusion it is noted tlat thecreticrAi.
problems have yet to be cleared up, in T~,articular those, related to
the concept of solving the relevant equations, classcs of func
tions met in the solutions, and the approximate m,Lthoas cf repre
senting these functions on Crids. The opinions expre.,~..,~ed in thJs
Card 4/5
12 8 C2/0C) I ""Oo I'Ir
0 ~2'1'
S
Differen?e methods for the D299/D30'5
article were formed during numer3us discuss.;Dnz '_n whi~h rathema
ticians in,,~_Iuding Keldysh, Gellfand, Babenkc, D yahenk~.'. ,:;ok pact,
There are 16 references: 13 Sovietbloc arid 3
U F113 e
(inluding 2 translations)~ The reference to the 21riglishlangU r
publication reads as follows: J Neumann, R Ri.Ii,,neyer, A riethod
for the numerical alculations of hydrQ,IynariJ.,. J, App.1,
Phy5, 1950, 21. no, 3, 232  237,
SUBMITTED: October 19, 1961
Card 5/5
_GCIDUNOV, SA, (Moskva); ZABRODIN, A.V. (~Wkva)
Difference schemes of secondorder accuracy for multidemensionai
problems. Zhur.vych.mat.i mat.fiz, 2 no.4'706708 JIAg 162.
CXMA 15;8)
(Difference equations)
42756
S/206 62/002/006/002/007
;,UTHOR: Godunov, S. '~. '"'Oscow)
T I TL P~ 'Uthod of ortnoeonalizution for solving of systems of
difference equations
PLAWDICAL: Zhurnal vychislitellnoy matematiki i matematicheskoy
fiziki, v, 2, no, 61 1962, 972982
TLAT; Difference equations of the.form
Lu.=(p, Ruv=IY,
A.,I.d~, + B.%u. = (n  1,2,....N), W
RuN
are considered. The solution algorithm is the followingt
A
YS(o) m 3;11/2(fs1/2 S_j /2z,(01)
"ard 1/3
C
S/208/62/002/006/002/007
Method of orthogonalization for ... B112/B166
Y A 0  1,2# ... 9k)
_112 1/2z.l
(1) (2) (k)
The vectors z 8 , Z s zs are obtained from the vectors y
(k) rtno~ naiization and normalization. The vectors z('),
YS by o. "o 0
(2) (S)
z z constitute a complete system of orthonormalized vectors satis
fying the condition LZ (j), . o 01,2,...,k). The vector Z(o)is perpendicular
(2) 0
M, (s)o `ilis the inhomogeneous et: ation Lz(o)
to z 2 z ana fui. U 0
The error of the proceso is esti;aatoi under certain reatrictions concerning
the difference scheme I is founI to be relatively small. The sub
ject of the paper resulted from a dioussion at the lAoskovskiy universitet
(Aosco,, University) in 1,)6,, initiated by N. S. Bakhvalov who suggested
~ method for orthogonalization of scalar equations of the type
~nun1 + bnu n+ cn unti . fn (cf. S. K. Godunov, V. S. Ryaben'kiy.
Card 2/3
5/2o8j62/002/006/002/007
Ylethod of orthogonalization for ... B112/B166
Vvedeniye v teoriyu raznostnykh skhem (Introduction into the theory of
difference schemes) M., k'izmatgiz [now printing) ).
SUBMITT10: May 30, 1962
f
Card 3/3
S/01'2/62/017i/003/002/002
B125/B104
A'U~ H 0 Godunov, S. K.
"Tl~ j;:
The proble,.! of a generalized solution in the theory of
quasili:near equations and in gas dynamics
I~E"'r'IODIC.'L: Usnekhi matematicheskikh nauk, v.
17, no. 300j), 1962,
14~158
This review deals with applications of the general solations of
UU  OU  0 to gas dynamics. S. L. Sobolev introduced this concept of a
7t Ux
,generalized solution in the oourse of an attempL to oliminnv~o tho
condition of s=oothness from the general solution u  f(x+t), The solution
to a Eiven differential equation can be generalized in variol~s ways. The
difficulties caused by the ambiguity of the generalized uolLtions to the
Cauc*.,y problem can be avoided by icposing additional limitations (e.6.
interral con,"'tions) on the generalized solutions. In thermodynamics, the
0 f
Drob.Lem of the integratIng factor seems to follow f"rom the conditions or
correctness of the differential equations. Attempts to discover
Card 1/2
The problem of a generalized
S/042//62/01 7/003/00~/002
B125/B104
relationshiDs between thermodynamics and the partial differential
aquations achieved success through a systematic classil'i'Cation' of the
different equations in mathematical physics. For systems that ire
symmetric in Friedrichs' sense of that word, a law similar to t*ae lair of
conservation of entropy can be derived. The LagrangeEuler variational
equation can be written as a system of three equations. In the one
dimensional case, the system, with dissipative terms is
uL /6t + dL1 /ax  (6/ax)b
qi qi ik(aqk/ax)
.here 1i 'D ikl; is symmetric and positively definite matrix. The c,rdinary
equations for solving (6) permit fine geometrical interpretations to be
nade. In !,as dynamics, a meaningful concept of generalized solutions
exists only for such equations of state as fulfil the conditions of
Bethe and 71eyl. The proof of the theorems of existence in very
difficult. There are 4 figures.
SUBL:ITTED: December 19, 1961
Curd 2/2
GODIJNOV) S.K.
Nonuniqueness for paraboli, srtems. Dok3.,,All SSSR 145 no3:499
500 Ja ;62. MIRA 15:7)
1. Pred6tavleno akWemikom I.G.Petrovsk"
(DIfferential equations)
I An m.. 91136ii 10 JMe
ANONICAL FORMS OF SYSTEMS OF LINEAR ORDINARY DIFFEREE'NCE rQU4
IONS WITH CONSTANT COEFFICIENTS (USSR)
~
Godunov S and V. S. Ryabenlkly. Zhurnal vychislit6llnoy mal emat;' i
matematicheskoy liziki, v. 3, no. 2, MarApr 1963, 211V22.
S 2 0 86:3 0 0 300 2001 Q 14
study is m ade of the systems of ordinary difference equations,
fk' n 0, + 1,...; k
Z j Vr.+ 1 n
_j
ZZ! i=
,4ith respect to functions vJ JvJ) of the argument n, vi ith t~e value v4 taken
n
ap a point havin'o, an abcissa n and ordinate j. Canonical forms of the bu6dlelof
matrices alk + PB, where a and 0 are parameters and A and B are matrices.~'are
a0alyzed. On the basis of this analysis, canonical for=s of the difference equation.
AU + 1Un+1 (2)
n a
'C~_rd 1/2
10 Jwne
CAYONIO5L POM 07 SYSM."M* (Cont,d] S/24/63/(A)3/()W/001/014
where U, and F., are vcctors, are presented, and their proper Ities studied,' ItAs
s,"Own "4 Vvitiout, its basic properties scalar sy:Htem JI), can be reduced
to a vector form (2) and that a unique solution of (1) corresponds to each solution of
(2). C, onditi onz for solving (1) are presented in the form o~f the*enis. The red,uc
tion of.the bou,.dary va.,.je pro4lem for (1) to that for (2) is investiggted. IL K)
215/2
GC)LjdNuV, S.K.; RIUFENIKI~. ~ .".
Spectral indications of the stability of boundary wilwi problems
for nonselfadjoint difference equations. Unp. mat. nauk 18
no.3:314 WJe 163. (MIRA 16tIO)
VASILIYEVI, O.F.; GODUNOV, S.K.; PRITVITS, N.A.z TEMNOYEVA, T.A.;
FRYAZIN6'vT,r'.L.,'"AiUdAfN, S.M.
Numerical method for calculating the propagation of long waves
in open river beds and its application to the flood prob:.em.
Dokl. AN SSSR 151 no.3t525527 JI 163. (MIRA 16.9)
1. Institut gidrodinamiki Sibirskogo otdoleniyrt AN SSSR.
Predstavleno akademikom P.Ya.Kochinoy.
ACCESSION NRs AP4037252 S/0200/64/004/()03/0473/GW
AUTHORSt Ademekayap 1. A. (Moscow); GoduftoVv So K. (906009)
TITLEs Kethod of spherical harmonion in the problein of critical parameters
SOURCEt Zhurnal V*ohislitellnoy matevatiki i matematichookoy fts:Udp To 4v to 3#
.1964P 473484
TOPIC TAGSt spherical harmonics; critical parameterp spherical re&ctorv atltigroup
approximation# reactor dimension
ABSTRACTt The authors study the problem of deternining aritio&l parameters of
spherical reactors in a multigroup approximation by the method of spherical har
monica. The problem for 2n harmonica and m group@ is reduced to a system of 2=
differential equations for 2 ma unknown functions yijr i m Ov 1j~o*o9(2n1)t j 
112jo.oyme The index i denotes the number of the hamonicy and j the number of
,the.group. The system of differential equations has the form
d
+ b, !!~:W + (T'Mi, t + 6Mil J) +yu P
dr dr
I. o.t,..., (2n  1); 1,2,...,
[Card
ACCESSION NRt AP4037252
Here
i+1 bi  1 0+00+2) 66 (2)
a, ~ __ 0 P Ti
121 ~+ 214 1
Vi is the velocity of neutron@ of the Jth groupq X to a parameter (time conAtant
.'of the system) and /a in density. Computing the variable y the components of
the vector y in 2ma dimensional space, the system can be rewritten as
P W;
+ Qy + XVy = pDy, (3)
W
T,
where P, Q, V, and D are matrices. The problem of finding critical parameters
be handled in the following manner. Considering atriotly given reactor dimensions,
find the least value of the parameter % for which (3) has a nontrivial solution
satisfying the given boundary conditions, or determine the least value of the
parameter,B p with IN  Op for which system (3) has a nontrivial solution in the
region [0,,pk3p satisfying the given boundary conditions (problem of critical
reactor dimensions), eto. The method proposed by the authors for solving this
problem io a trial method* Given suoceesiY917 the values of the parameter being
determined (p or N ), one solven system ()) and each time computes &om variable
d  "residual" whiohp roughly speakingp shows how mah one boun"ry oonditi*n is
not satisfied when the other is "tiefiode Trials are mad* until# for the abassm.
Card 2/3
ACCESSION NRs AP4037252
value of the paramterp the residma is practically eqma to seroo "A great dftl of.
the work in setting up the Programs (without which this p*W oouM act have beem.
written) was done by I@ Fe Nzarovas" Orig. art. bass 12 foramlame
ASSOCUTION& nano
SUBICUMS 13M463 BATS ACQs 09Junk JWL%1 00
So=$ MA NORM Uffs 0" &IM I (X*
Card 3/3
Q66 E1NT(l)/FEC(k)~/EWA(h)
ACC NR, Ap6oi5631 SOURCE ODDS: UR1011131661=10091003810038
INVENTOR: Kovarskiy, B. 1. Go&Wov. V. I.
.,'ORG: none
v:
or Wt
'Resonance vayemelif  the tMF ranpo' Clime:
ek~
dbri i IL'
~80URCZ.' 17,0 ten y pronVablenWye obraztsy, tovarVye =ski.,; 9,' 1966, 38
on
Otopic as waveguide element, waveguide frequency, wavegmIde ~raovmissi
:ABSTRACT: 'The UHF resonance wavemeter shown in the figure consists of a tudable
cylindrical H wave resonator exicted by a waveguide splitter.vhizb encloses the
A
1.4
1  resonator fo,r preliminexy
tuning; 2  waveguide splitter,
3  detector;h  coupling dia
A.A 4  rescmietor for pr~
VievB phraga; 5
A . . I
cise tuning.
1/2 UDC.* 621317763
~c rd
L 2562o66
ACC NR, AP6015631
resonator, and a resonance indicator working in conj~uqction with a detector.. High
measurement accuracy is achieved by coupling the H011 respnittor to atiother cylindrical,
resonator by means of a transverse diaphragm. The second resonator aperatell on the
Ho d wave& Its frequency may be independently tuned. Drig. art. hati:1 figtite. (BD]
W
M CODE, 14,'0g/ SUMB DATE, 28sep63/ ATD PRESS:
Itard 2/2
GODUNP_V.,,.Iurly.I!Uplgn,0&4; OACHEIV, Aleksey Gavrilovjc~,
ULASHNIKOV, AnatolAy Fedorovich; KOLESNIKOV, A:LLk3,ndr
Sergeyevich; DEVOCHKIN, MI.., red.
(The greenbelt; practices in the establishment of park
foreet plantations and orchards around Volgograd) Zele
nos kolitso; opyt sozdaniia lesoparkovykh nasazhdenil i
sadov vokrug Volgograda. Volgograd, NizhneVolzhskoe
knizhnoe izdvo., 1964. 100 p. (H I RA 18. 3)
GODUNDVA, G.S.; MECOYE'A, I.I. (Leningrad, Tylenirla, d.',
Subtroc"han"eric osteotomy with subsequent skeletal traction in
coxa vara in children. Ortop., traviri. .1 protess. 25 no.5:50 My
164. (MIRA 184)
1. Iz Detskogo ortopedicheskogo instituta iperil G.1,'N.nera (dir. 
prof. M.N.Goncharova), Leningrad.
GOMNOVA, G.S., mladshiy nauchnyy sotivinik (Leningrad, Nevskiy prospokt, d.210,
kv. 7)
Agerelated indications "or surgical treatment of cang
I gerital syndactyly
of the hand. Ortop., tram i protez. 25 no.812731 Ag 164.
(MIRA 1814)
1. 1z Detskogo ortopedichnskogo Instittita Imenk Ttirnern (dir.  prof.
M.N.Goncharova), Lentngrad.
GQDIII~QV, K. :~'.
v
27812. Godunova, X. 11. Sort i plodorodiye pochvy (SortolFTytnriye o,imo.,r
yarovoy pshenitoi). Selektalya i eemenoyodptvo, 1949, No. 9, s.
SO: Letopis I ZhurrAlInykh Statey, Vol. '17, 1949
USSR/Soil Science. Tillage. Melioration. Erosion J5
Abs Jour Ref Zhur  BiOlj, No 10.9 1958, No 43875
Author Lnrlunnyn TC?I
Inst Not Given
Title Freliminary Results of a Study of T.S. Mailtsev's metilol of
Soil TillinG Made an Variety Testing Plots
Orig Pub Inform. byul. Gos. komis, po sortoiapyt. kulltur pri
Mve a.kh. SSSR, 1956, No 4, 38
Abstract A survey of the use of deep nonterraced plowing and surface
tillaGe made in 1955 at 200 varic~y testing plots and in
the kolkhozes of the taiC;a and steppe zones of the Euroijean
part of the USSRp Western Siberia) North Kazakhstan and the
Northern Caucasus. AccordinC; to the findinC.;s of 9 variety
plots in Western Siberia, deep nonterraced plowing on cherno
zem soils provided an increase in the productivity of surmer
wheat of from 1 to 2.8 centners per ha. The application of
deep nonterraced plowinG with an uplifting of the fa..l plow
Card 1/3
33
USsil/soil Science. Tillage, Melioratiow Erosion J5
Abs jour Rof Zhur  Biol., No 10, 1958) No 43875
land was advantageous to sumier Vheats According to 'J30
data of the variety testing Plots along the VolgMt dc,)p non
terracinr, plowinG in the fall provided an increase in the
y1eld in 4 of the 5 tests. Deep nonterracinC plovinG Of the
perennial grass layer loverad the yield from 0.4 to 4.8 centners
per ha. The use of mitiple diskinG without plowLnG the
field after the perennial LTasses did not sbcW positive re
suits. The suraner wheat yield in NosltovskaYa Oblast with this
til,lage was 4*5 centners per ha. lower than by plovint"~ thc
field with a plow hiavinG, a colter* The deep nonterracc0
workinG of a fallow for winter wheat Oa P0dZ0lic soils haa
no advantace over ordinary plairinG. In the Ukraine and the
North Caucasus the use of deep nonterracin4; after plowed
crops also did not increase the harvest. III these rayans
positive results were gotten frum the surface workin3 of the
soil. Deep soil treatment provided o. sumer wheat) :iate and
comi yield boost* Positive results were obtainea frou the
application of surface treatment on weedfree land for the
Card 2/3
GODUNGVA, K.N... kand.sel'skokhozyaystvennykh nauk
Contribution of new cultivation practices to production.
Zemledelie 23 no.90341 3 161. (NIRA 14 ' 12)
1. Goskomissiya po nortiosipytaniyu eel 'okokho,,yaysfven:1y!Ji
kul I tur.
(AgTiculturo)
GODUNOVA, K.N., kand.sell8kokhozyaystveOnykh nauk
Seeding rates and dates for winter wheat in nonChernozem
&Teas. Zemledelie 24 no.7:2731 n 62. (YLIRA 15: 12)
1. Gosudarstvanr
fya komissiya po sortoispytaniyu sellskokho
zyaystvermykh kulltur.
(wheat) (sowing)
GbDUNC,VA, K.N., kand.sellskokhozyaystvennykh nauk; KNOPOV, r.v.
Effectiveness of manuresoil composts in the experiments of state
variety testing stations. Zemledelie 25 no.2t4952 F 163.
(MIRA 160)
(compost)
KIABUIIOVSEIY, Ye.I.; BALANDIN, A.A.: WDUNOVA, L.F.
Ghromatographic separation of menthol. Izv. jUl SSSR Otd.khIm.nauk
no.12:22432244 D '61. MRA 14.11)
1. Institut organicheskoy khimli im. N.D.Zelinskogo AN SSSR.
(Menthol)
KLABUNOVSKIY, Te.l.; BALANDIN, A.A.; GODUNOVA, L.F.
Inversion of 1menthone. Izv.AN SSSR Otd.khix'lnauY no.5.886890
.My 163. (MIRA 16:8)
1. Institut organicheqol khimii im. N.D.Zelinskogo AN SSSR.
(Ma%thanoneOptical properties)
BADALOV) S.T.; BASITOVA, S.M.; GODUNOVA, L.1
Distribution of rhenium in molybdenites in Central Asia..
Geokhimiia no.9:813817 162. (MIRA 15:1:)
1. Institute of Geology, Academy of Sciences of the Uzink
Soviet Socialist Republic., Tashkent wW Institute of Chumistry)
Academy of Sciences of the Tadzhik Soviet Socialist Repiblic.,
Dushanbe.
(Soviet Central AsiaRhenium)
(Soviet Central AsiaMo*demim ores)
W/11~4 o/twoo.
A&914209 1411 ATS002793
910VKM j44PjkbJBW _42n firi go, gnk ffsnjjjt.~ it, 1~'. ua~ Ii'~ 19
soveshchanT a. Hasleavo ladw*o Nuaks, 196 2314.95f
(Rhenium); trudy
TOPIP Tiftt rhanium stionium datermittat tong, Matybdeatto sulfide!
gtt8fij colorimetry.0 molyWenum precipitation.
BSTRAM The authors stu4iilo the ~ optimal 'condWo f04 1 ~Ilw 6VOL, iolft~. id
an list
A
Mra
rh 'nium In sulfide minerals, walybdenites; in ps;,iicular, isl4or iiepora~ in oil
molybdenum by coprecipItation with Iron hydroxide.  A sol;blt oil lini~klitibdonit r vLl.h
itnown contents of rhenium werestudied. The inol*bdaftita to'
114914i'' V014 i4caw led ?it
Aittric acid, the excess of the latter was driven off wt 0 ilt, maybd a t*% F&S
copra* Lpits tad with iron hydroxide by &mean"* 'the ptecio"I'll 414t1trIZ&ild,, and
rbenium was deteraLmd, eolortmetriently In the sautwd,4,111 i J 041inAte CJ31 tP t. 
In addition, the authors. developed & teelftnique for delter0i Onj 'I lboinlum in W;hexI
sulfide mineral's e'uch ma chalcopyriteap. pyrites, sphaldrit li, r
cuntent of which is muich lower. This w" done 0 41 :
log t~ "~lo VLI;h 1
Cor.s 1/2
GODIMOVA, N.K. [Hodunova, N.K.], kand_med.nuak; PRAVDIIIA, L.I.
liffoct of exercise therapy on external breathing, in pro,,nant wonen
and now mothers with cardiovascular diseases. Ped,alru3h. i gin.
20 no.2:4650 158. (MIRA 13:1)
1. Otdel vnutrenney patologii beremennykh (zav.  kand.mod.nauk
N.A. Panchenko) Ukrainskogo InstItuta klinicheolkoy maditsiny (direkbor 
prof. A.L. Mikhnerv).
(RESPIRATION) WARDIOVASGULAR SYSTA.MDIS119ASES)
 : .: I . I. .I .., ~ '. , " , .  Z.
.1 . ~ A !" . ; ";~ ,
, . ) . ~ . I I I
 I " : I ',, I " , , " T ",
,  ., I I .. .1 . %~ ; ., . . . i
LEVIN, V.I.; OODUNDVAP YO.K. (Moskva)
Ganeralisation of Carlson's inequality,, Mat, sbor, 67
no#4i643446 AS 165, (MIRA 1818)
SOURCE CODEt
'63/C20/006
AUTHORt Godunova, Ye. K.
ORGt none
TITM Integral inequality with an arbitrary convex function
SOURCE: Uspekhi, matematicheskikh nauko v. 20, no. 6, 19650 6667
TOPIC TAGSs integration, function
ABSTRILCT; 'in a Of
One of thq equautie
ca
d*Vg (m) dy) dx I and arbitraj
.7
(xy) of the product of tulo, v=b?os rd
nonnegative function K 19willot'le: N. Dunfo,
!'and Jo Schwartzp LineyWe operatory (Mnear Operutorv).p P age 576)t
ca
K(zp)g(m)dV1pdZ < NP op (V) dg, (2)
Nz= X(I)VIdl.
Card 1/4
ACC NRt AM19390
,Th* present note proves an inequality which is 'a generallUtiCA Of the Utte'r:
,the power function is replaced by a monotone convex functiam.
.defined on
Theorems Let $D(u)p and K(u), and f(u) be nonnagative functions,
Positive semimkis ( a p (u) in in*roasings co tinuousp and convex;
(0) = 0; rc(u) and I? lu')r"brelong to L (0j oD ); lot 9 (u) 4a w N o Tho
PA
'the folloidug ex&4 inequality is Y;llds
tP Y (s))
Y
In order to PMO'IMRU&litr Wo it,is sufficient to use the pro e;r of U4
jp/ ds
0
P Its P di
01
c6ngiths order of intilgrat
Card 2 /4_
W. I Ile oil IN I III l" 1"Wi[NIM1111 It.. 1112111 'AIIIIIHM IIIH
ACC NRt AP6019390
Ids K (zy) IV (I (y)) d dif a"
xX (xy) I (v) dir
C*
dz dy M dy.
The result is a strict inequality: the sign of equalityin is posoible
: only if f (x) = constp which case is ruled out since _VAf f7a to
:1, (0" 00).  7 belon,
To prove the exactness of inequality (3), replace f(x) in it with
i Ar S